# Questions tagged [lie-superalgebras]

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67
questions

**6**

votes

**2**answers

211 views

### Lie powers of a graded vector space and Klyachko's theorem

Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...

**6**

votes

**0**answers

78 views

### Reference request: superconformal algebras and representations

I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...

**3**

votes

**1**answer

138 views

### Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...

**2**

votes

**1**answer

113 views

### Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...

**9**

votes

**2**answers

171 views

### Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...

**1**

vote

**0**answers

37 views

### is this the correct universal property of free Lie superalgebras?

Consider a $\Bbb Z_2$ graded set $A$.
Universal property of free Lie superalgebra $FLS(A)$: Let $\mathfrak g$ be a Lie superalgebra and let $\Phi: A \to \mathfrak g$ be a set map which preserves the $\...

**1**

vote

**0**answers

41 views

### Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras.
A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...

**1**

vote

**2**answers

303 views

### Doubt in the Serre relation and the odd/even roots of a Lie superalgebra

Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.
A real matrix $A=(a_{ij})_{i,j\in I}$ is ...

**2**

votes

**1**answer

78 views

### Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...

**1**

vote

**0**answers

69 views

### Quadratic Lie superalgebras

Let $(L,.)$ be a Lie superalgebra endowed with an even supersymmetric non-degenerate and invariant bilinear form $B$ (i.e $(L,.,B)$ is a quadratic Lie superalgebra). If we have the equality $B(x,y.z)=(...

**7**

votes

**0**answers

104 views

### Infinitesimal description of homogeneous supermanifolds

Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...

**10**

votes

**3**answers

746 views

### Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...

**1**

vote

**0**answers

35 views

### Cartan integers for Lie superalgebras

Let $\mathfrak g$ be a basic classical simple Lie superalgebra (BCSLSA in short) with Cartan matrix $A$ of some simple system $\Pi$. Then $A$ satisfies the conditions that
1) $\frac{2a_{ij}}{a_{ii}} \...

**2**

votes

**0**answers

52 views

### Super characters in the theory of Lie superalgebras

Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...

**2**

votes

**1**answer

211 views

### Category of typical representations for Lie superalgebras

I am interested in studying the category of typical representations over basic classical simple Lie superalgebras. In particular, I want to know
1) is this category semisimple (character determines ...

**2**

votes

**0**answers

46 views

### Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...

**2**

votes

**0**answers

52 views

### Dynkin diagram of Basic classical simple Lie superalgebras

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...

**4**

votes

**1**answer

205 views

### Typical and atypical modules for Lie superalgebras

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...

**4**

votes

**2**answers

311 views

### Lie super algebra presentation of the Kähler identities

For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities:
\begin{align*}
[\partial,L] = 0, ...

**1**

vote

**0**answers

70 views

### GKO construction for (Super-)Virasoro algebras

I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...

**2**

votes

**1**answer

126 views

### super Lyndon words

Lyndon words form a basis for Free Lie algebras.
In this direction, I need the reference for the super Lyndon words for free Lie superalgebras.
Given the definition of super Lyndon words, how to ...

**2**

votes

**1**answer

101 views

### Lyndon basis of free Lie superalgebras

Lyndon basis for Free Lie algebras is well known in the literature.
My question is,
what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of ...

**4**

votes

**0**answers

64 views

### Reference Request: Representation Theory of Real Lie Superalgebras

Are there some references for the representation theory real lie superalgebras, specifically of $psu(1,1|2)$, and $u(1,1|2)$?

**5**

votes

**2**answers

434 views

### Serre relations for Lie Superalgebras

Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory.
I have the following ...

**2**

votes

**0**answers

89 views

### Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...

**10**

votes

**1**answer

462 views

### Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...

**2**

votes

**0**answers

64 views

### odd positive roots of basic classical Lie superalgebras

In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras.
...

**2**

votes

**0**answers

180 views

### action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...

**1**

vote

**1**answer

130 views

### finite dimensional modules are highest weight modules [closed]

Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector.
My feeling is, since $e_i$'s ...

**1**

vote

**0**answers

66 views

### how the borel subalgebras of p(n) look like except the standard one?

I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this.
I am trying to do the proof of the Proposition ...

**7**

votes

**2**answers

531 views

### Character formula for Lie superalgebras

The Weyl character formula and the denominator identity play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$
Can you suggest ...

**2**

votes

**1**answer

123 views

### $P(1)$ strange type classical Lie superalgebras

I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson.
The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...

**0**

votes

**1**answer

109 views

### left ideals in Lie super algebras

Let $\mathfrak g$ be a Lie superalgebra.
If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?
For me ...

**2**

votes

**3**answers

361 views

### Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...

**2**

votes

**0**answers

121 views

### Two definitions of super-Virasoro algebra

Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...

**8**

votes

**1**answer

486 views

### Vanishing of H-cohomology

This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence ...

**2**

votes

**1**answer

215 views

### Dual Coxeter Number for Superalgberas

I am looking for a reference that gives the definition and has summarized the dual Coxeter number for superalgebras, especially for $\mathfrak{u}(m|n)$ (the Lie algebra of unitary supergroup $U(m|n)$)....

**2**

votes

**0**answers

57 views

### Action of orthosymplectic group $SpO(d+1|d)$ on $PGL(d+1|d)$

Suppose $d$ is odd, and consider the super vector space $\mathbb{C}^{d+1|d}$ and its super projectivization $\mathbb{P}^{d|d}$.
In the case $d=1$, a paper of Witten's, concerning the moduli space of ...

**2**

votes

**1**answer

216 views

### Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ...

**3**

votes

**1**answer

193 views

### Endomorphismensatz for Lie superalgebras

For semisimple complex Lie algebras there is Soergel's Endomorphismensatz
$$C = \operatorname{End}(P(w_0)) \cong \mathbf C[\mathfrak h]/\mathbf C[\mathfrak h]^W$$
for $w_0$ the longest element in ...

**3**

votes

**0**answers

35 views

### Is there a name for Lie superalgebras which are generated by the odd subspace?

Every Lie superalgebra $\mathfrak{g} = \mathfrak{g}_{\bar 0} \oplus \mathfrak{g}_{\bar 1}$ has a canonical ideal $\mathfrak{k} = [\mathfrak{g}_{\bar 1}, \mathfrak{g}_{\bar 1}] \oplus \mathfrak{g}_{\...

**4**

votes

**1**answer

367 views

### Definition of orthosymplectic supergroups

I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard.
The first version of the ...

**3**

votes

**1**answer

127 views

### Reference request: coordinate ring of $OSP(2p|n)$

In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.
Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are ...

**1**

vote

**1**answer

237 views

### Super version of Poisson brackets of tensor products

Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule).
Super version of the product of two tensor products is
\...

**9**

votes

**0**answers

166 views

### Super-extensions of the Poincaré Lie algebra

For $(\mathfrak{g},[-,-])$ an ordinary Lie algebra let me say that a super-extension of it (maybe not the best terminology) is a super-Lie algebra $(\mathfrak{s}, [-,-]_{\mathfrak{s}})$ whose bosonic ...

**3**

votes

**0**answers

134 views

### Classical Yang-Baxter equation for Lie algebras and Lie superalgebras

The classical Yang-Baxter equation is
\begin{align}
[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1)
\end{align}
What are the differences between this equation in the case of Lie ...

**0**

votes

**1**answer

87 views

### References request: vector representations of Lie superalgebras

Are there some references of fundamental representations of Lie superalgebras (in particular for the Lie superalgebra $sl(m|n)$? Thank you very much.

**0**

votes

**2**answers

547 views

### Two definitions of the super Jacobi identity

In this paper, page 149, the super Jacobi identity is given by
\begin{align}
J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0.
\end{align}
But in this article, ...

**4**

votes

**1**answer

122 views

### Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra

I have a solution (a $R$ matrix) of the Yang-Baxter equation,
\begin{equation}
R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1})
\end{equation}
that probably ...

**3**

votes

**0**answers

122 views

### Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra

Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...